Computing Coefficients Of Modular Forms

This thesis is about computational aspects of modular forms and Galois representa-tions attached to them.In the book[23], Couveignes, Edixhoven et al. described algorithms for computingcoefcients of modular forms for the group SL2(Z), and Bruin[12]generalized the methodto modular forms for the congruence subgroups of the form Γ1(n). Their methods leadto polynomial time algorithms for computing coefcients of modular forms. However,efcient ways to implement the algorithms and explicit complexity analysis are still be-ing studied. Working with complex number field, Bosman’s explicit calculations showthe power of these new methods, see[23]. As one of the applications, he largely improvedthe known result on Lehmer’s nonvanishing conjecture for Ramanujan’s tau function. Inthis thesis, we present a reduction mod p algorithm for computing coefcients of modularforms. Instead of using Brill-Noether’s algorithm, we work with the function field of themodular curve, using He’s algorithm to make computations in the Jacobian of modularcurve. Our totally algebraic algorithm is clearly structured with an explicit complexityanalysis. Using the algorithm, we[71]first completed the calculation of projective repre-sentation associated to mod31and improved the numerical verification of Lehmer’snonvanishing conjecture. We[17]also improved the reduction mod p algorithm by usingmodular curve XH(N) and developed algorithm for computing equations for XH(N) withmoduli interpretation. We use these new algorithms to compute Galois representationsassociated to discriminant modular forms for values of that are significantly higher thanin prior works. This thesis will include detailed description of these algorithms as well.